$L^p$ regularity of least gradient functions
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- by Wojciech Górny PDF
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Abstract:
It is shown that in the anisotropic least gradient problem on an open bounded set $\Omega \subset \mathbb {R}^N$ with Lipschitz boundary, given boundary data $f \in L^p(\partial \Omega )$ the solutions lie in $L^{\frac {Np}{N-1}}(\Omega )$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.References
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Additional Information
- Wojciech Górny
- Affiliation: Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland
- Email: w.gorny@mimuw.edu.pl
- Received by editor(s): April 23, 2019
- Received by editor(s) in revised form: April 24, 2019, and November 25, 2019
- Published electronically: April 9, 2020
- Additional Notes: This work was supported in part by the research project no. 2017/27/N/ST1/02418, “Anisotropic least gradient problem”, funded by the National Science Centre, Poland.
- Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3009-3019
- MSC (2010): Primary 35J20, 35J25, 35J75, 35J92
- DOI: https://doi.org/10.1090/proc/15031
- MathSciNet review: 4099787